We prove some detailed quantitative stability results for the
contact set and the solution of the classical obstacle problem in
()
under perturbations of the obstacle function, which is also equivalent to studying the
variation of the equilibrium measure in classical potential theory under a
perturbation of the external field.
To do so, working in the setting of the whole space, we examine the evolution of the free
boundary
corresponding to the boundary of the contact set for a family of obstacle functions
. Assuming
that
is
in
and that the initial free
boundary
is regular,
we prove that
is twice
differentiable in
in a
small neighborhood of
.
Moreover, we show that the “normal velocity” and the “normal acceleration” of
are
respectively
and
scalar
fields on
.
This is accomplished by deriving equations for this velocity and acceleration and
studying the regularity of their solutions via single- and double-layer estimates from
potential theory.
